After doing some more searching, I may have found the root of the problem. It’s wonderful what a mere write up for a forum does to your problem solving.
Honest Mobile is a mobile virtual network operator (MVNO). Not sure if this is relevant at all, but what is definitely relevant is that Three UK is switching off their 3G. The things which is almost certainly relevant is when Three UK says:
Calls – your device must be 4G Calling – also known as VoLTE – compatible. Some older 4G devices don’t support 4G Calling.
So I go to my “Network and Internet” → “SIMs” → “Honest” and 4G Calling is switched off. I switch it on and suddenly I get 4G+ and full coverage.
Now this is a solution, so I am going to mark it as such, but I would still appreciate if someone comments on why this is necessary.